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Self-testing nonprojective quantum measurements in prepare-and-measure experiments

TAVAKOLI, Armin, et al.

Abstract

Self-testing represents the strongest form of certification of a quantum system. Here we investigate theoretically and experimentally the question of self-testing non-projective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterised measurement device implements a desired non-projective positive-operator-valued-measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension, which we argue is natural for this problem since any measurement can be made projective by artificially increasing the Hilbert space dimension. We develop methods for (i) robustly self-testing extremal qubit POVMs (which feature either three or four outcomes), and (ii) certify that an uncharacterised qubit measurement is non-projective, or even a genuine four-outcome POVM. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment.

Specifically, we show that our experimental data implies that the implemented measurements are very close to certain ideal three and four outcome qubit POVMs, and hence non-projective. In the [...]

TAVAKOLI, Armin, et al . Self-testing nonprojective quantum measurements in

prepare-and-measure experiments. Science Advances , 2020, vol. 6, no. 16, p. eaaw6664

DOI : 10.1126/sciadv.aaw6664

Available at:

http://archive-ouverte.unige.ch/unige:138223

Disclaimer: layout of this document may differ from the published version.

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Armin Tavakoli,1Massimiliano Smania,2 Tamás Vértesi,3 Nicolas Brunner,1and Mohamed Bourennane2

1Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

2Department of Physics, Stockholm University, S-10691 Stockholm, Sweden

3Institute for Nuclear Research, Hungarian Academy of Sciences, P.O. Box 51, 4001 Debrecen, Hungary Self-testing represents the strongest form of certification of a quantum system. Here we investigate theo- retically and experimentally the question of self-testing non-projective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterised measurement device implements a desired non-projective positive-operator-valued-measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension, which we argue is natural for this problem since any measurement can be made projective by artificially increasing the Hilbert space dimension. We develop methods for (i) robustly self-testing extremal qubit POVMs (which feature either three or four outcomes), and (ii) certify that an unchar- acterised qubit measurement is non-projective, or even a genuine four-outcome POVM. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data implies that the implemented measurements are very close to certain ideal three and four outcome qubit POVMs, and hence non-projective. In the latter case, the data certifies a genuine four-outcome qubit POVM. Our results open interesting perspective for strong “black-box” certification of quantum devices.

I. INTRODUCTION

Measurements in quantum theory were initially represented by complete sets of orthogonal projectors on a Hilbert space.

Such measurements are standard in a multitude of applica- tions. Nevertheless, in a modern understanding of quantum theory, measurements are described by positive-operator val- ued measures (POVMs), i.e., a set of positive semi-definite op- erators summing to identity. POVMs are the most general no- tion of a quantum measurement; all projective measurements indeed are POVMs, but not all POVMs need be projective.

Non-projective measurements are widely useful in both conceptual and applied aspects of quantum theory, as well as in quantum information processing. In several practically motivated tasks, they present concrete advantages over pro- jective measurements. Non-projective measurements enhance estimation and tomography of quantum states [1,2], as well as entanglement detection [3]. Furthermore, they allow for unambiguous state discrimination of non-orthogonal states [4–6], which would be impossible with projective measure- ments. They have also found applications in quantum cryp- tography [7,8] and randomness generation [9]. In addition, non-projective measurements can be used to maximally vio- late particular Bell inequalities [10] (assuming a bound on the Hilbert space dimension), a fact that has been applied to im- prove randomness extraction beyond what is achievable with projective measurements [11,12].

In view of their diverse and growing applicability, it is im- portant to develop tools for certifying and characterising non- projective measurements under minimal assumptions. The strongest possible form of certification involves a “black-box”

scenario, where the quantum devices are a priori uncharac- terised. Astonishingly, it is possible in certain cases to com- pletely characterise both the quantum state and the measure- ments based only on observed data, which is referred to as

“self-testing” [13]. A well-known example is that the max- imal violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [14] implies (self-tests) a maximally entangled two-qubit state, and pairs of anti-commuting local projective

measurements [15–17]. Self-testing can also be made robust to noise [18–20].

However, for the purpose of characterising non-projective measurements in the black-box scenario, methods based on Bell inequalities encounter a challenge. Due to Neumark’s theorem, every non-projective measurement can be recast as a projective measurement in a larger Hilbert space. That is, any non-projective measurement on a given system is equiva- lent to projective measurement applied to the joint state of the system and an ancilla of a suitable dimension, see e.g. [21].

Since in the Bell scenario one usually considers no restriction on Hilbert space dimension, it is non-trivial to characterise a non-projective measurement based on a Bell inequality. While this is possible in theory (in the absence of noise) [11], it ap- pears challenging in the more realistic scenario where the ex- periment features imperfections. A possible way to circum- vent this problem is to consider a Bell scenario with quantum systems of bounded Hilbert space dimension. In particular, Refs [12,22] recently reported the experimental certification of a non-projective measurement in a Bell experiment assum- ing qubits. However, these experiments do not represent self- tests, as they certify the non-projective character of a measure- ment, but not how it relates to a specific target POVM.

Here we investigate the problem of self-testing non- projective measurements. We follow a different approach, by considering a prepare-and-measure scenario instead of a Bell scenario. We argue that this approach is well suited to the problem, and offers a natural framework for certifying and characterising non-projective measurements. Indeed, as the problem almost naturally involves an upper bound on the Hilbert space dimension of the quantum systems (as discussed above), the prepare-and-measure scenario is sufficient. In this case, as opposed to Bell experiments, there is no need to in- volve distant observers and entangled states. From a practi- cal point of view this makes a very significant difference, as prepare-and-measure experiments are notably simpler to im- plement [23–29]. Moreover, prepare-and-measure scenarios are easier to analyse theoretically, which allows us to develop self-testing methods that are versatile and highly robust to

arXiv:1811.12712v2 [quant-ph] 13 Dec 2018

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noise.

In the first part of the paper, we present methods for char- acterising non-projective measurements. Firstly, we present a method for self-testing a targeted non-projective measure- ment in noiseless scenarios. Secondly, since noiseless statis- tics never occur in practice, we present methods for inferring a lower bound on the closeness of the uncharacterised mea- surement and a given target POVM, based on the observed noisy statistics; specifically, we lower-bound the worst-case fidelity between the real measurement and the ideal target one.

Thirdly, we introduce a method for determining whether the observed statistics could have arisen from some (unknown) projective measurements. If not, the measurement is certified as non-projective. These methods have two-fold relevance.

On the one hand, they enable foundational insights to phys- ical inference of non-projective measurements in the black- box scenario. On the other hand, they provide tools for as- sessing and certifying the quality of an experimental setup.

We demonstrate the practicality of these self-testing methods in two experiments. In the first, we target a symmetric in- formationally complete (SIC) qubit POVM and demonstrate an estimated 98% worst-case fidelity in the black box sce- nario. Additionally, our data certifies a genuine four-outcome qubit POVM. In the second experiment, we target a symmetric three-outcome qubit POVM and certify a worst-case fidelity of at least96%. Finally, we discuss some open questions.

II. THE SELF-TESTING PROBLEM, THE SCENARIO AND OVERVIEW OF RESULTS

Self-testing is the task of characterising a quantum system based only on observed data, i.e., black-box tomography. In other words, it is about gaining knowledge of the physical properties of initially unknown states and/or measurements present in an experiment by studying the correlations observed in the laboratory.

In this work, we focus on prepare-and-measure scenarios.

They differentiate themselves from Bell scenarios in two im- portant ways. Firstly, prepare-and-measure scenarios involve communicating observers and thus no space-like separation.

Secondly, they do not involve entanglement, whereas Bell sce- narios do. Prepare-and-measure scenarios can generally be modelled by two separated parties, Alice and Bob, who re- ceive random inputsxandyrespectively. Alice prepares and sends a quantum stateρxto Bob who performs a measurement ywith outcomeb, represented by a POVM{Myb}bwith

Myb≥0 and X

b

Myb=11 ∀y. (1) This generates a probability distribution

p(b|x, y) = tr ρxMyb

. (2)

In order to make the problem non-trivial, an assumption on Alice’s preparations is required; otherwise Alice could simply sendxto Bob and any probability distributionp(b|x, y)would be achievable. The assumption we consider in this work is

that Alice’s preparations, i.e. the set of statesρx, can be rep- resented in Hilbert space of given dimensiond. By choosing d < |x|, we prevent Alice from communicating all informa- tion about her inputxto Bob. Importantly, there exist dis- tributions obtained from quantum systems of a dimensiond that cannot be simulated classically, see e.g. [30,31]. That is, no strategy in which Alice communicates a classicald- valued message to Bob can possibly reproduce the observed data. Such distributions that cannot be classically simulated are candidates for self-testing considerations.

The problem of self-testing consists in characterising the set of states{ρx}and/or the set of measurements{Myb}based only on the distributionp(b|x, y). This characterisation can usually be done only up to a unitary transformation and possi- bly a relabelling. In a recent work [32], methods were pre- sented for self-testing sets of pure quantum states, as well as sets of projective measurements in the qubit case. These were subsequently extended to higher dimensional systems in Ref. [33].

Formally, a self-test can be made via awitness, which is a linear function of the probability distributionp(b|x, y):

A[p(b|x, y)] = X

x,y,b

αxybp(b|x, y) (3) whereαxyb are real coefficients. Moreover, given a witness, one can determine its maximal witness valueAQ achievable under quantum distributions (2) in a bounded Hilbert space.

The witness can then be used for self-testing a set of quan- tum states and/or measurements, whenever there is a unique combination of states and/or measurements that achievesAQ. Then, it is clear that when the observed distributionp(b|x, y) leads toAQ, a specific set of states and/or measurements is identified (up to a simple class of transformations). A nec- essary condition for a witness to be useful for self-testing is that, for a given dimensiond, quantum systems outperform classical ones; if not, several strategies would generally be compatible with the data; see Refs [23,24,30,34] for exam- ples of such witnesses. In SectionIII A, we present a method for constructing witnesses whose maximal value can self-test a targeted non-projective qubit measurementMtarget.

Next we turn to robust self-tests, i.e., self-tests that can be applied even when the statistics is not ideal causing the wit- ness value to be less thanAQ. Indeed, this is fundamental in order to make our methods applicable in practice, as any realistic experiment is prone to noise. The influence of noise makes it impossible to perfectly pinpoint the states and mea- surements. This motivates the following question. Given an observation of a witness valueA < AQ, how close are the states and measurements to the ideal ones, i.e. those that would have been perfectly self-tested if we had observedA= AQ. In Section.III B, we develop methods for robustly self- testing non-projective qubit measurements by lower-bounding the fidelity between the implemented measurement and the ideal one. A tight robust self-testing would give the fidelity between the measurement that is most distant from the ideal one, and that could have generated a witness valueA<AQ.

Whereas robust self-testing represents a quantitative physi- cal inference, it is also relevant to consider a more qualitative

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inference. Based on the witnesses we develop for self-testing, we show how to certify that the uncharacterised measurement is non-projective. In section III C we determine the largest value of our witness that is compatible with qubit projec- tive measurements. When observing a larger value, the non- projective character of the measurement is certified. In a simi- lar spirit, we determine a bound on our witness above which a genuine four-outcome (non-projective) qubit measurement is certified.

An overview of all the self-testing methods developed in this work are illustrated in Fig.1. The methods will be applied in sectionIVto self-test particularly relevant non-projective qubit measurements. For these examples, we will demonstrate the usefulness of our methods by implementing them in a pho- tonic experiment. Specifically, our experimental data implies that the implemented measurements are very close to certain ideal three- and four- outcome qubit POVMs, and hence are non-projective. In the latter case, the data certifies a genuine four-outcome qubit POVM.

III. CERTIFICATION AND CHARACTERISATION OF NON-PROJECTIVE QUBIT MEASUREMENTS This section presents methods for certification and char- acterisation of non-projective measurements in prepare-and- measure scenarios both with noiseless and noisy statistics.

The focus will be on qubit systems. Therefore, we begin by summarising the properties of qubit POVMs.

A positive operator-valued measure (POVM) with O out- comes is a set of operators {Ei}Oi=1 with the property that Ei ≥0and thatP

iEi =11. In the case of qubits,Eican be represented on the Bloch sphere as

Eii(11+~ni·~σ), (4) where~ni (with|~ni| ≤ 1) is the Bloch vector, λi ≥ 0, and

~

σ = (σx, σy, σz)are the Pauli matrices. Positivity and nor-

FIG. 1: Graphical overview of the self-testing methods and steps presented in sectionIII.

malisation imply that

O

X

i=1

λi= 1 and

O

X

i=1

λi~ni= 0. (5)

The set of POVMs is convex, and a POVM is called ex- tremal if it cannot be decomposed as a convex mixture of other POVMs. For qubits, extremal POVMs have eitherO= 2,3,4 outcomes [35]. In the caseO= 2, extremal POVMs are sim- ply projective, whereas forO = 3andO = 4they are non- projective; an extremal three-outcome qubit POVM has three unit Bloch vectors in a plane, and an extremal four-outcome qubit POVM has four unit Bloch vectors of which no choice of three are in the same plane [35]. An extremal qubit POVM is therefore characterised by its Bloch vectors. As the statistics of non-extremal POVMs can always be simulated by stochas- tically implementing extremal POVMs, it is clear that only extremal POVMs can be self-tested.

A. Self-testing non-projective measurements: noiseless case Consider a target extremal non-projective qubit POVM Mtarget, withO = 3or O = 4outcomes, for which we as- sociate the outcomebto the unit Bloch vector~vb. Our goal is now to construct a witnessAsuch that its maximal value self- testsMtarget. The method consists in two steps summarised in Fig2.

Step 1.First we construct a simpler witnessA0featuringO preparations, i.e. Alice hasO inputs. Bob receives an input y = 1, . . . , Y and provides a binary outcome. The goal of this simpler witness is to self-test a particular relation among the prepared states|ψxi. Specifically, we would like to certify that their unit Bloch vectors~uxpoint in opposite direction (on the Bloch sphere) to those of the target POVMMtarget, i.e.

~ux=−~vxforx= 1, ..., O. Let us define A0= X

x,y,b

cxybP(b|x, y), (6)

with real coefficientscxybchosen such that the maximal value A0Qof the witness for qubits self-tests the desired set of pre- pared states{|ψxi}(up to a global unitary and relabellings).

In general, we believe that it is always possible to find such a self-test by considering enough inputs for Bob, corresponding to well-chosen projective measurements, and suitable coeffi- cientscxyb; see Ref.[32] for examples. Furthermore, note that one could also in principle have more thanOpreparations for Alice, and then self-test thatOof them have the desired rela- tion toMtarget. In addition, we remark that the construction of an adequate witnessA0is not unique in general.

Step 2. We construct our final witnessAfromA0. Specif- ically, we supply Bob with one additional measurement set- ting calledpovm. This setting corresponds to a measurement withOoutcomes. Since the intention is to self-test the mea- surement corresponding to this setting asMtarget, we associate

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FIG. 2: Method for self-testing a targeted non-projective qubit mea- surement by exploiting simpler self-tests of preparations. Step 1:

tailor scenario and witness such that a maximalA0self-tests Alice’s preparations to have Bloch vectors that are anti-aligned with those of the target measurement. Step 2: Add an extra setting to Bob and modify the witness to self-test the target non-projective mea- surement.

the settingpovmtoOoutcomes. We define A=A0−k

O

X

x=1

P(b=x|x,povm), (7) for some positive constant k. A maximal witness value AQ = A0Q now implies that the settingpovmcorresponds toMtarget(up to a unitary and relabellings). This is because a maximal witness value implies that (i) the set of prepared states {|ψxi}have Bloch vectors anti-aligned with those of Mtarget, and (ii) P(b = x|x,povm) = 0 for allx, hence the Bloch vectors of the settingpovmare of unit length and aligned with those ofMtarget. Moreover, as a qubit POVM is characterised by its Bloch vectors, we see thatMtargetis the only POVM that can attain the maximal witness value AQ. Therefore we obtain a self-test of the target POVMMtarget.

In sectionIV, we will apply this method to self-test sym- metric qubit POVMs with three and four outcomes.

B. Robust self-testing of non-projective measurements No experiment can achieve the noiseless conditions needed to obtain exactly a maximal value of A. Therefore, it is paramount to discuss the case when a non-maximal value of Ais observed. We will show that in this case, one can never- theless make a statement about how close the uncharacterised

measurementEperformed in the lab (corresponding to the set- tingpovm) is to the target POVMMtarget.

In order to address this question, we must first define a measure of closeness between two measurements. A natu- ral and frequently used distance measure in quantum infor- mation is the fidelity, F, between two operators. Inspired by previous works [19,32,36,37], we consider a measure of closeness amounting to the best possible weighted aver- age fidelity between the extremal qubit target POVM elements Mtarget = {Mi}and the actual POVM elementsE = {Ei}.

That is, we allow for a quantum extraction channelΛto be applied to the actual POVM. This extraction channel must be unital (i.e., identity preserving) in order to map a POVM to an- other POVM. Clearly, we look for the best possible extraction channel. We thus define the quantity

F E,Mtarget

= max

Λ

1 2

O

X

i=1

tr (Λ[Ei]Mi)

tr (Mi) . (8) Since the target measurement is extremal, the POVM ele- ments are proportional to rank-one projectors; Mi ∝ Pi. Due to (4) we can write Λ[Ei] = λi(11+~ni·~σ) subject to the constraints (5). By evaluating (8) we find thatF = 1/2 + 1/2P

iλitr (Pi~ni·~σi) ≤1. To saturate the inequal- ity, each Bloch vector~nimust be of unit length, i.e. |~ni|= 1, and aligned with the Bloch vector ofPi. Hence,MiandΛ[Ei] are both proportional to the same rank-one projector. Since a POVM with Bloch vectors of unit length is fully characterised, i.e. all coefficientsλiare fixed by the conditions (5), this im- plies thatMi = Λ[Ei]. Thus, a maximal fidelity ofF = 1 is uniquely achieved when the actual POVM is equal to the target measurement1.

In general, a non-maximal value of the witnessAcan arise from many different possible choices of states and measure- ments. We denote byS(A)the set of allO-outcome POVMs that are compatible with a given observed valueA. Our goal is now to find a lower-bound on the average fidelityFthat holds foreverymeasurementE0 ∈S(A). Therefore, the quantity of interest is theworst-caseaverage fidelity:

F(A) = min

E0∈S(A)F E0,Mtarget

. (9)

Calculating this quantity, or even lower-bounding it, is typ- ically a non-trivial problem even in the simplest case. We proceed with presenting two methods for this task.

We remark that the definition (8), given for qubits, could potentially be extended to higher-dimensional systems (re- placing the factor1/2by1/d). This could work for POVMs where all elements are proportional to rank-one projectors.

However, the latter are only a strict subset of general extremal POVMs. Finding a more general figure of merit is thus an interesting open question.

1We note that this would not necessarily be the case when the POVMMiis not extremal. However, as mentioned above, for the purpose of self-testing it is enough to focus on the case whereMiis extremal.

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Robust self-testing with the swap-method.— A lower- bound on the worst-case average fidelity can be obtained via semidefinite programming [40]. The method combines the so-called swap-method [41,42], introduced for self-testing in the Bell scenario, and the hierarchy of dimensionally bounded quantum correlations [38]. Such adaptations of the swap- method to prepare-and-measure scenarios were introduced in [32] to self-test pure state and projective measurements. In AppendixAwe outline the details of how the swap-method is adapted to robustly self-test non-projective measurements.

This method benefits from being applicable in a variety of sce- narios and for returning rigorous lower-bounds onF. Never- theless, it suffers from two drawbacks. Firstly, the method only overcomes the fact that self-tests are valid up to a global unitary, but not that they may be valid up to relabellings.

Thus, it is only useful for target measurements that are self- tested up to a unitary. Secondly, while rarely producing tight bounds onF, the computational requirements scale rapidly with the number of inputs, the number of outputs, and the chosen level of the hierarchy. In SectionIV, we will show that the method can be efficiently applied for robustly self-testing a three-outcome qubit POVM.

Numerically approximating robust self-testing.—In order to also address cases in which self-tests are valid up to both a unitary transformation and relabellings, we can estimateF based on random sampling. The approximation method ben- efits from being straightforward and broadly useful, while it suffers from the fact that it merely estimates the value ofFin- stead of providing a strict lower-bound. The key feature is that the minimisation appearing in Eq. (9) is replaced by a minimi- sation taken over data obtained from many random samples of the settingpovm. We detail this method in AppendixB, and apply it to an example in Section.IV.

C. Certification methods for non-projective measurements Whereas robust self-testing considers quantitative aspects of physical inference from noisy data, it is important to also consider the qualitative inference. An important qualitative statement is to prove that the uncharacterised measurement is non-projective, or more generally, that it cannot be simulated by projective measurements. Indeed, it is known that when POVMs are sufficiently noisy, they become perfectly simula- ble via projective measurements [21,43,44]. The witnesses we construct can address this question. We will see that when- ever the observed value of the witnessAis sufficiently large, one can certify that the settingpovmnecessarily corresponds to some non-projective measurement, and could not have been simulated via projective measurements. Specifically we de- rive an upper-bound on A for projective measurements (or convex combination of them). The violation of such a bound thus certifies a non-projective measurement, or more precisely a genuine three (or four) outcome POVM. At the end of this subsection, we also show how to certify a genuine four out- come POVM.

A projective qubit measurement has binary outcomes, and can therefore be represented by an observableM ≡M0−M1

whereMiis the measurement operator corresponding to out- comei= 0,1. Let us consider the case where theO-outcome measurementpovmis projective. One may assign two out- comes to rank-one projectors and the rest to trivial zero op- erators. Note that it is enough here to consider these cases, as the witnessAis linear in terms of the measurement op- erators. Projectors can thus be assigned in three (O = 3) or six (O = 4) different ways, of which the optimal in- stance must be chosen. Let the outcomes in the optimal in- stance beo0|povmando1|povm, and associate the observable Mpovm≡MY+1=Mpovmo0|povm−Mpovmo1|povm. The witness (7) can be written as

A=C(k) +X

x

trh

ρxL(k)x ({My})i

, (10) whereC(k)is a constant, andL(k)x ({My})is a linear com- bination of the observables{M1, . . . , MY+1}. Note that that L(k)x ({My})does not depend on the indexybut on the col- lection of observables. Using the Cauchy-Schwarz inequality for operators we obtain,

A ≤C(k) +X

x

r trh

ρxL(k)x ({My})2i

. (11) Due to projectivity, we have My = ~ny · ~σ, where ~ny is of unit length. Using {Mk, Ml} = 2~nk ·~nl11, one finds L(k)x ({My})2 =t(k)x ({~ny})11, for some functiontwhich is a weighted sum of scalar products of the Bloch vectors of the observables. Consequently, in order to boundAunder all pro- jective measurements, we have

A

Proj

≤ C(k) + max

{~ny}

X

x

q

t(k)x ({~ny})≡ B(k). (12) Thus, B(k) bounds the value of A for projective measure- ments. The evaluation of this bound only depends on Bob’s Bloch vectors and is further simplified by their parameterisa- tion in terms of two angles2.

Moreover, when targeting a four-outcome qubit POVM, we consider also a finer form of qualitative characterisation by considering whetherAcan be simulated by the settingpovm being some three-outcome POVM. If not, the measurement is certified as a genuine four-outcome measurement. This amounts to bounding the value ofAachievable under any two- or three-outcome qubit POVM and then observing a violation of that bound. For this purpose, one may employ the hierarchy of dimensionally bounded quantum correlations [38] which can be used to upper boundAunder three-outcome POVMs

3. To obtain tight bounds, one may need a reasonably high hi-

2The effort needed to evaluate the bound depends on the chosen prepare- and-measure scenario. Typically, considering scenarios with some sym- metry properties is beneficial.

3The hierarchy is built on projective measurements. This obstacle can be overcome by embedding Alice’s preparations in a larger Hilbert space with the dimension chosen such that three-outcome POVMs can be re-cast as projective measurement following Neumark’s theorem.

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erarchy level which can be efficiently implemented using the methods of Ref. [39].

Next, in Section.IV, we will apply the outlined methods to specific non-projective measurements and experimentally demonstrate the certification of both non-projective and gen- uine four-outcome measurements.

IV. RELEVANT EXAMPLES AND THEIR EXPERIMENTAL REALISATION

In the above, we have discussed methods for self-testing a target non-projective measurement. Here we put these methods in practice in a photonic experiment. We imple- ment three- and four-outcome symmetric qubit POVMs, with Bloch vectors forming a star (trine-POVM) and a tetrahe- dron (SIC-POVM) respectively. In the first case we certify a non-projective measurement, and apply our methods for ro- bust self-testing, demonstrating worst-case average fidelity of at least96% compared to an ideal trine-POVM. In the sec- ond case, we certify a genuine four-outcome qubit POVM, and demonstrate worst-case average fidelity of approximately 98%with respect to an ideal SIC-POVM. Below we will first present the setup common to both experiments and then con- sider each example separately by first applying the methods of sectionIIIto obtain adequate witnesses, and then present the corresponding experimental realisation.

Experimental setup –In the experiment, the qubit states are encoded in the polarisation degree of freedom of a single pho- ton, with the convention of|Hi ≡ |0iand|Vi ≡ |1i. The setup is depicted in Fig. 3. Alice’s station includes a her- alded single photon source where femto-second laser pulses at 390 nm are converted into pairs of photons at 780 nm, through type-I spontaneous parametric down-conversion in two orthogonally oriented beta-barium borate crystals (BBO).

Photon pairs go through 3 nm spectral filters (SF), and are then coupled into two single-mode fibres (SMF) for spatial mode filtering. The idler photon is sent to the trigger APD detector (T), and heralds the presence of a signal photon. The latter is then emitted again into free-space, and undergoes Alice’s state preparation, consisting of a fixed linear polariser (POL), aλ/4(QWP) and aλ/2(HWP) wave-plate.

Upon preparing the required qubit state, Alice forwards the signal photon to Bob’s measurement station, where it goes through a double-path Sagnac interferometer, each path of which contains an HWP. The interferometer mixes the polar- isation degree of freedom with path, effectively enabling Bob to perform either projective or non-projective measurements in the original polarisation Hilbert space where the qubit was prepared, thanks to the two polarisation analysers at the out- puts. Each of these consists in a phase plate (PP), a HWP and (in output 1) a QWP, a polarising beam-splitter and two single- photon detectors. Outputs from all detectors (T, D1-D4) are sent to a coincidence unit connected to a computer.

All measurements were performed with heralded photon rates of approximately 1 × 104 counts per second, while each setting was measured for 500 seconds. The quality of state preparation and measurement can be estimated by

preparing states |Hi, |+i = (|Hi+|Vi)/√

2 and |Ri = (|Hi+i|Vi)/√

2, and measuring them in the Pauli basesσz, σxandσyrespectively. The three visibilities obtained in our setup with this characterisation measurement were:

Vσz = (99.91±0.02)%

Vσx = (99.31±0.01)% (13) Vσy = (99.23±0.02)%.

While the almost optimalVσzis a direct consequence of the high extinction ratios of the PBSs used, the lower visibilities in the interference bases are mainly due to the double-path Sagnac interferometer, which showed a visibility of around 99.4%, therefore effectively bounding from above the results we can achieve in the experiments.

A. Example I: the qubit SIC-POVM

We begin by illustrating the self-testing methods for a fre- quently used non-projective measurement, namely the qubit SIC-POVM, which we denoteMSIC. This measurement has four outcomes and its four unit Bloch vectors{~vb}b form a regular tetrahedron on the Bloch sphere, with weightsλb = 1/4. Such a regular tetrahedron construction can be achieved via two different labellings of the four outcomes that are not equivalent under unitary transformations. Up to a unitary transformation, each such SIC-POVM can be written with Bloch vectors

~v1= [1,1,1]/√

3 ~v2= [1,−1,−1]/√ 3

~v3= [−1,1,−1]/√

3 ~v4= [−1,−1,1]/√

3, (14) and the set of Bloch vectors{−~vl}lrespectively.

Noiseless self-test.— We find a prepare-and-measure sce- nario for self-testingMSIC. Following step 1 in sectionIII A, we introduce a prepare-and-measure scenario in which Alice has four preparations, x ∈ {1,2,3,4}, and Bob has three binary-outcome measurements,y ∈ {1,2,3}. The witness is chosen as

A0SIC= 1 12

X

x,y

P(b=Sx,y|x, y), (15)

where S1,y = [0,0,0], S2,y = [0,1,1], S3,y = [1,0,1], and S4,y = [1,1,0]. The maximal value, A0SIC = 1/2 1 + 1/√

3

, can be achieved by Alice preparing her four states forming a regular tetrahedron, e.g., with the Bloch vectors in Eq. (14), and Bob performing the measurements σx, σy andσz. The four vectors experimentally prepared by Alice, as obtained by state tomography, are reported in Fig. 4(left). In AppendixC, we prove the maximal witness value and show that it self-tests that Alice’s preparations in- deed must form a regular tetrahedron on the Bloch sphere.

By step 2 in sectionIII A, we supply Bob with an additional four-outcome measurementpovm, and consider the modified

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FIG. 3: Experimental setup. More details, including labelling, can be found in the main text.

FIG. 4: States prepared by Alice for the SIC-POVM (left) and trine- POVM (right) experiments. The vectors are obtained from qubit state tomography.

witness ASIC= 1

12 X

x,y

P(b=Sx,y|x, y)−k

4

X

x=1

P(b=x|x,povm).

(16) Thus, we conclude that ASIC = 1/2(1 + 1/√

3) self-tests MSIC.

We note that there also exists other prepare-and-measure scenarios fulfilling the requirements of step 1. For exam- ple, one may achieve the desired self-test using the so-called 3→1random access code whose self-testing properties were considered in Ref. [32]. However, this prepare-and-measure scenario requires more preparations than the one presented here.

Robust self-test –Next, we consider the worst-case fidelity (given in Eq. (9)) of the measurement corresponding to the settingpovmwithMSIC. Since the self-test ofMSICis valid up to a relabelling as well as a collective unitary, we cannot use the swap-method to lower bound F. Instead, we em- ploy the numerical approximation method (see Appendix B for details). Figure5displays roughly3×105optimal pairs

(ASIC, F)each evaluated from a randomly sampled measure- ment for the setting povm. The evaluation was done for k = 1/5(which, as will soon be shown, turns out to be the most noise-resilient choice of k). We see that the minimal sampled fidelity as a function ofASICdescribes a curve which constitutes the approximation ofF.

Certifying non-projective and genuine four-outcome POVMs.—Finally, we derive a tight bound valid for all qubit projective measurements on the value ofASIC. Due to the symmetries ofASIC, we can without loss of generality let the non-trivial (non-zero measurement operator) outcomes of the measurementpovmbe the outcomesb = 1,2. Hence, we define the observableMpovm ≡ M4 =Mpovm1 −Mpovm2 . Then, we follow the steps outlined in sectionIII C. First, we re-writeASIC in the form (10). We findC(k) = (1−2k)/2 and

L(k)x=0,1({My}) = 1 24

1,(−1)x,(−1)x,(−1)x+112k

·M~ L(k)x=2,3({My}) = 1

24

−1,(−1)x,(−1)x+1,0

·M ,~ (17) whereM~ = [M1, M2, M3, M4], withMy =~ny·~σ. After ap- plying the Cauchy-Schwarz inequality, we obtain a cumber- some expression of the form of Eq. (11). In order to evaluate its maximal value (following Eq. (12)), we use the following concavity inequality: √

r+√ s ≤ p

2(r+s)forr, s ≥ 0, with equality if and only if r = s. Apply this inequality twice to the expression (12), first to the two terms associated tox= 0,1, and then to the two terms associated tox= 2,3.

After a simple optimisation over~n3and denotingx=~n1·~n2, one arrives at

ASIC≤1−2k

2 +

√2 24

√6−4x

+

√ 2 24

q

2rk+ 4x+ 48k√ 2√

1 +x≡fk(x),

(9)

FIG. 5: Numerical approximation of the worst-case fidelity of the unknown measurement (settingpovm) with the qubit SIC-POVM by roughly3×105random three- and four-outcome POVM samples for which the optimal values of(A, F)were calculated. The figure also displays the critical limits onASIC and F for projective and three-outcome POVMs respectively, as well as the experimentally measured values.

whererk = 3 + 144k2. This bound is valid for a particu- lar value of x. In order to hold for all projective measure- ments, we simply maximisefk(x)overx. This requires only an optimisation in a single real variablex ∈ [−1,1]which is straightforward. The optimal choice is denotedx. Set- ting B(k) = fk(x), we have ASIC ≤ B(k) for all pro- jective measurements. Although the expressions involved are cumbersome, the analysis is simple and straightforward.

We have considered the tightness of the projective bound for k ∈ {1/100,2/100, . . . ,1}by numerically optimising ASIC under unit-trace measurements (which includes all rank-one projective measurements). In all cases, we saturate the bound B(k)up to machine precision with a projective measurement.

Furthermore, we have also considered boundingASIC un- der three-outcome qubit POVMs using the hierarchy of di- mensionally bounded quantum correlations (as described in section III C). In our implementation of [38], we have em- bedded the qubit preparations into a three-dimensional Hilbert space, and optimisedASIC under projective measurements of the only existing non-trivial rank-combination. The relax- ation level involved some monomials from both the second and third level, and the size of the moment matrix was126.

This was done for allk∈ {1/100,2/100. . . ,1}, and each up- per bound was saturated up to numerical precision using lower bounds numerically obtained via semidefinite programs.

In order to study the robustness of both the non-projective and the genuine four-outcome certification, we have consid- ered the critical visibility of the system needed when exposed to white noise. This is modelled by the preparations taking the formρx(v) = vρx+ (1−v)11/2wherev ∈ [0,1]is the visibility. Denoting byArandthe witness value obtained from the optimal measurements performed on the maximally mixed state, the critical visibility for violating some given boundB is

vcrit(k) = B(k)− Arand+k

AQ− Arand+k . (18) We have applied this toASICwithB(k)corresponding to the bounds on projective and three-outcome measurements re-

FIG. 6: Critical visibilities for certifying a non-projective measure- ment and a genuine four-outcome measurement respectively, in the prepare-and-measure scenario (16) targeting the qubit SIC POVM.

TABLE I: Wave plate settings for the experimental setup (as appear- ing in Fig.3) for the experiment based on the SIC-POVM. All angles are in degrees. The preparation Bloch vectors~niof Alice point to the vertices of a regular tetrahedron.

Alice

state Bloch vector HWP QWP

~n1 20.07 17.63

~n2 24.93 -17.63

~n3 -24.93 17.63

~n4 -20.07 -17.63

Bob

setting H1 H2 HWP1 QWP1 HWP2 PP1 PP2

1 0 0 22.5 0 – 0 –

2 0 0 0 45 – 0 –

3 0 0 0 0 – 0 –

povm 13.68 31.32 0 45 22.5 45 135

spectively. The corresponding critical visibilities appear in Figure6. In both cases, we find that the largest amount of noise is tolerated fork= 1/5, corresponding tovcrit= 0.970 andvcrit= 0.990respectively.

Experimental result –Wave-plate settings (referring to Fig.

3) for Alice’s prepared states in Eq. (14) and Bob’s measure- mentsσx, σy, σz, and the four-outcome SIC-POVM anti- aligned to the vectors in Eq. (14), are reported in Tab.I.

Optimally choosingk = 1/5, the measured value of the witness as compared to the relevant bounds is

ASIC projective

≤ 0.77383-outcome≤ 0.7836

qubit

≤ 0.7887.

ALabSIC= 0.78514±5×10−5stat±1.0×10−4syst. (19) The statistical error originates from Poissonian statistics and the systematic error originates from the precision of the wave plate settings. More details about the errors are discussed in AppendixE.

We observe a substantial violation of both the projective measurement and the three-outcome measurement bounds.

Thus, we can certify that Bob’s measurementpovmis a gen- uine four-outcome qubit POVM. Furthermore, as illustrated

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by the results in Fig5, we certify approximately a98%worst- case fidelity with the qubit SIC-POVM.

B. Example II: the qubit trine-POVM

We consider a second example in which the target POVM is the so-called trine-POVM. This measurement has three out- comes and its Bloch vectors form an equilateral triangle on a disk of the Bloch sphere, withλl = 1/3. The Bloch vectors are hence defined by

~

v1= [0,0,−1] ~v2= 1 2

h−√ 3,0,1i

~ v3= 1

2 h√

3,0,1i . (20) Noiseless self-test –We introduce a prepare-and-measure scenario in which Alice has three inputs x ∈ {1,2,3}, and Bob has two binary-outcome measurements labelled byy ∈ {1,2}, and consider the witness

A0tri= X

x,y,b

Tx,y(−1)bP(b|x, y), (21)

whereTx,1 = [1,1,−1]andTx,2 = √ 3,−√

3,0

. In Ap- pendixC, we show that its maximal value is A0tri = 5and that this value implies that Alice’s three preparations form an equilateral triangle on the Bloch sphere. Then, we add an ad- ditional inputpovmfor Bob and consider the witness Atri= X

x,y,b

Tx,y(−1)bP(b|x, y)−k

3

X

x=1

P(b=x|x,povm), (22) for somek > 0. Then,Atri = 5self-tests the settingpovm as the trine-POVM up to a unitary.

Robust self-test –We now turn to considering its robust self- testing properties, i.e., lower-bounding the worst-case fidelity of the unknown measurement (setting povm) with the tar- get measurement for a given value of Atri. Since the above self-test is achieved only up to unitary transformations, we may find rigorous lower bounds on the worst-case fidelity F using semidefinite programming. In accordance with sec- tion III B, we have performed the swap-operation on Bob’s side and used the hierarchy of finite-dimensional correlations to lower boundF 4. In addition, for sake of comparison, we have implemented the numerical approximation method for robust self-testing to estimate the accuracy of the bound ob- tained via the swap-method. The results are displayed in Fig- ure7. Although the bound obtained from the swap-method is most likely not tight, it will prove sufficient for the practical purpose of experimentally certifying the targeted POVM with high accuracy.

4The hierarchy level was an intermediate level containing some higher-order moments corresponding to an SDP matrix of size105.

FIG. 7: Lower bound onF(Atri)fork= 1obtained from the swap method, together with roughly 3000 points(Atri, F)obtained via the numerical approximation method. This is displayed next to the ex- perimentally achieved results.

Finally, we have also self-tested the trine-POVM in a differ- ent prepare-and-measure scenario (see AppendixC). In Ap- pendixD, we use this prepare-and-measure scenario to derive a tight bound on projective measurements by evaluating the right-hand-side of (12).

Experimental realisation –The witness in Eq. (22) is max- imised if Alice’s three Bloch vectors point to the vertices of an equilateral triangle on a disk of the Bloch sphere. We take that disk to be thexz-plane (see Fig. 4(right)), taking~ti =−~vi

(from (20)), and Bob performs one of three measurementsσz, σxand the three-outcome POVM with vectors anti-aligned to Alice’s states. In contrast to the previous experiment, the out- put 2 of Bob’s measurement station only consists of one de- tector (D3), and no wave-plate or PBS (see Fig.3). The wave- plate settings corresponding to the above states and measure- ments are reported in Tab.II.

TABLE II: Wave plate settings for the experimental setup (as ap- pearing in Fig.3) for the experiment based on the trine-POVM. All angles are in degrees. The preparation Bloch vectors~tiof Alice point to the vertices of an equilateral triangle.

Alice

state Bloch vector HWP QWP

~t1 0 0

~t2 30 0

~t3 -30 0

Bob

setting H1 H2 HWP1 QWP1 HWP2 PP1 PP2

1 0 0 0 0 – 0 –

2 0 0 22.5 0 – 0 –

povm 0 27.37 22.5 0 – 0 –

Using these settings, we have obtained the experimentally measured value ofAtri as a function ofk. Since we aim to demonstrate a large worst-case fidelity with the trine-POVM, we have computed the lower bound onF(Atri)for many dif- ferent values ofkand found that choosingk= 1leads to the optimal result. The corresponding experimentally measured

(11)

witness is Atri(k= 1)

projective

≤ 4.89165

qubit

≤ 5 (23)

ALabtri (k= 1) = 4.96587±7×10−4stat±1.7×10−3syst. (24) This data-point, and its relation to the worst-case fidelity of the lab measurement with the targeted POVM is depicted in Fig.7. FromALabtri , we infer a closeness of at least96%. This can be compared to the largest possible fidelity between a pro- jective measurement and the trine-POVM, which is straight- forwardly found to be(2+√

3)/4≈0.933. However, as is in- dicated by the results of the sampling based numerical approx- imation method for robust self-testing (presented in Fig.7), a better bound ofF may allow us to rigorously infer a worst- case fidelity of at least97.3%.

Furthermore, we have considered the possibility of the ex- perimental data certifying a non-projective qubit measure- ment. However, to this end, we found that another choice ofk is optimal with respect to the witness value that is achievable under projective measurements. We found that the optimal choice isk ≈ 4.5. The corresponding experimentally mea- sured value becomes

Atri(k= 4.5)

projective

≤ 4.71139

qubit

≤ 5

ALabtri (k= 4.5) = 4.93613±5×10−5stat±1.0×10−4syst. (25) We conclude that our experimental data certifies a non- projective qubit measurement.

V. CONCLUSIONS

We investigated the problem of self-testing non-projective measurements. We argued that a prepare-and-measure sce- nario with an upper bound on the Hilbert space dimension represents a natural framework for investigating this problem.

We considered both the qualitative certification of a measure- ment being non-projective and/or genuine four-outcome, as well as a quantitative characterisation in terms of worst-case fidelity to a given target POVM. We demonstrate the practi- cal relevance of these methods in two experiments in which we both certify a genuine four-outcome POVM, and infer a high worst-case fidelity with respect to target symmetric qubit POVMs.

It would be interesting to overcome the limitation of the swap-method and develop a rigorous robust self-testing method for general four-outcome qubit POVMs. Also extend- ing these methods to high-dimensional POVMs would be rel- evant since there exist extremal non-projective measurements that feature the same number of outcomes as projective mea- surements (contrary to the qubit case). Moreover, it would be interesting to investigate self-testing of non-projective mea- surements using different assumptions as in our work. One could consider for instance prepare-and-measure scenarios with a bound on the entropy [46], the overlap between the pre- pared states [9] or their mean energy [47]. Finally, one may ask whether it would be possible to robustly self-test a non- projective measurement in the fully device-independent case, i.e. returning to the Bell scenario without any assumption on the dimension.

Note added.—During the completion of this manuscript, we became aware of an independent work [48] discussing the certification of qubit POVMs.

VI. ACKNOWLEDGEMENTS

We thank J˛edrzej Kaniewski for insightful comments. This work was supported by the Swiss national science foundation (Starting grant DIAQ, NCCR-QSIT), the Swedish research council, and Knut and Alice Wallenberg Foundation. T.V. is supported by the National Research, Development and Inno- vation Office NKFIH (Grant Nos. K111734 and KH125096).

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Appendix A: Swap-method for robust self-testing of non-projective measurements

We outline the swap-method for robust self-testing of non-projective qubit measurements. Bob is supplied with trusted ancillary qubits

¯

ρb=|ψ¯biB’hψ¯b|, (A1)

where B’ denotes the ancillary system of Bob. The key idea is that Bob will swapρ¯binto his measurement box via an operation depending on his uncharacterised measurements, and then perform the de facto measurement (determined by the inputsy and povm) on the swapped system.

Let us recall our figure of merit from Eq. (8), whereMtarget={Mb}b is the target measurement. Without loss of generality, we can let every POVM elementMbbe proportional to a rank-1 projector with a proportionality factor2λb(see Eq. (5) for the definition ofλb)5. The ancillary qubits are initialised such thatMb = 2λbρ¯bcorresponding to the ideal case in which a noiseless self-test can be made. We may write the worst-case fidelity in Eq. (9) as

F(A) = min

{Mb}∈S(A)max

Λ

1 2

X

b

tr [Λ[Mb] ¯ρb], (A2)

Next, we obtain a lower bound on this quantity by using the swap method adapted to POVMs in the prepare-and-measure scenario (e.g. see Ref. [32] for self-testing state preparations in these setups). To this end,ρ¯bin Eq. (A1) is swapped into Bob’s box. Hence, in Eq. (A2) we replaceρ¯bby

ρSWAPb = trB’

h

SWAP(ρb⊗ρ¯b)SWAPi

, (A3)

where SWAP=U V Uis the swap operator with

U =11⊗ |0iB’h0|+B1⊗ |1iB’h1|, V =1 +B0

2 ⊗11B’+1−B0

2 ⊗(σx)B’, (A4)

whereB0andB1are for Bob’s qubit observables corresponding to projective two-outcome measurements. Note that in the ideal case, one hasB0zandB1x, and then SWAP defines the two-qubit swap operator

SWAP=

1

X

i,j=0

|iihj| ⊗ |jiB’hi|. (A5)

However, if Bob’sB0andB1measurements which give rise to the maximal witness valueA0 in Eq. (6) are not equivalent up to unitary rotations toB0zandB1x, we can still construct the swap operator. We can always rotate Bob’s local bases such thatB1xandB0= sinθσz+ cosθσxfor someθangle. In order to derive the optimal operator used to construct the swap operator, we can naively setBz= (B0−cosθB1)/cosθ. However, this operator needs not be unitary in general when it is expressed in terms of the moments. Still there always exist a unitaryB2such thatB2Bz≥0. We then useB2in the construction of the swap instead ofB0, which ensures the unitarity of the swap operator. More on this so-called localising matrix approach can be found in Refs. [41,42].

By replacingρ¯bwith (A3) in Eq. (A2), we obtain F(A) = min

{Mb}∈S(A)max

Λ

X

b

1 2trh

Λ[Mb]⊗11SWAP(ρb⊗ρ¯b)SWAPi .

We can further write SWAP= (1/2)P1

i,j=0sij⊗ |iiB’B’hj|, wheresijare some linear combinations of polynomial expressions inB0,B1and11, whose exact forms can be found in Appendix H of Ref. [32]. Using the above decomposition for SWAP one arrives at

F(A) = min

{Mb}∈S(A)

1 8

X

b 1

X

i,j,k=0

tr (Ri,j,k,b)hj|¯ρb|ki, (A6)

5 Note that in Sec.IV, the presented case studies will focus onλb= 1/d, wheredis the number of outcomes.

(14)

whereRi,j,k,b =Mbsijρbski. Note that the channelΛmaps measurements to measurements, therefore we safely ignored it in the above expression.

Lower bounds valid for any quantum realisation of the (A6) can be obtained via the hierarchy of Ref. [38]. This is a finite dimensional SDP method using a random sampling approach to construct a feasible moment matrix Γ with entriesΓi,j = tr

QiQj

, whereQidefines a sequence of operators such thatΓcontains all variablesRi,j,k,band also contains all variables building upA. Then the solution of the following SDP provides a lower bound toF(A), whereAstands for a fixed value of A:

F(A)≥ 1 32min

Γ

X

b 1

X

i,j,k=0

tr (Ri,j,k,b)hj|ρ¯b|ki (A7) such that Γ≥0 and A ≥ A.

Note that during the random sampling, we generate quantum states|ψbiinC2along with traceless qubit observablesByinC2. However, in order to simulate a randomd-outcome measurementMinC2, one has to generate random projective measurements in a larger space, and then embed the preparations|ψbiand observablesByin this larger space. Consequently, a feasible moment matrixΓis constructed from these quantum states and measurements [38].

Appendix B: Approximation method for robust self-testing of non-projective measurements

In this appendix, we detail a simple procedure for estimatingFin (9) for robust self-testing. The key idea is to replace the minimisation in (9) by instead sampling a large number of elements from the relevant set of POVMs.

First, sample a random extremald-outcome qubit POVM corresponding to the settingpovm. Numerically compute the max- imal value ofASICover theOpreparations and theY binary-outcome measurements. For qubits, this can be straightforwardly achieved using Bloch sphere parameterisation and the fact that all variable measurements are optimally taken as projective. An alternative method is to use semidefinite programs in see-saw [45]. Then, optimiseF over the unital extraction channelΛ. A simple way of obtaining a lower bound onFis to relaxΛto a unitary, which benefits from a simple Bloch sphere parameterisa- tion. Perform this optimisation separately with respect to each of the possible relabellings of the target measurement that are not equivalent under unitary transformations, and then choose the largest one. In this manner, one obtains an optimised pair(A, F) for the given measurement sample for the settingpovm. Repeating this procedure many times, and collect a large sample of pairs(A, F). Then, one can estimate the minimisation over the setS(A)appearing in Eq. (9) by instead, for eachAtaken in a suitably small interval, choosing the smallestF among the obtained pairs(A, F). In order to have a good approximation of F, one requires a reasonably large number of samples as well as a reasonably unbiased manner of sampling the measurement povm.

Appendix C: Self-testing non-projective measurements with noiseless statistics

In this Appendix, we exemplify the method for self-testing a targeted non-projective measurement in various cases. First, we focus on the four-outcome qubit SIC-POVM discussed in the main text. It is defined up to a unitary and relabellings, by Bloch vectors~vr0r1 = 1/√

3 [(−1)r0,(−1)r1,(−1)r0+r1], and weightsλr0r1 = 1/4, forr0, r1 ∈ {0,1}. Then, we consider the three-outcome trine-POVM also discussed in the main text. It is defined up to a unitary, by Bloch vectors~v1 = [1,0,0],

~ v2 =

−1,0,√ 3

/2and~v3 =

−1,0,−√ 3

/2, and weightsλr = 1/3, forr ∈ {1,2,3}. For these two cases, we first use the corresponding prepare-and-measure scenarios described in the main text and prove the desired self-tests. Then, we prove self-testing of the trine-POVM in a different type of prepare-and-measure scenario exhibiting practical symmetries.

1. Self-testing the SIC-POVM

In the main text, an example of a prepare-and-measure scenario is given in which a maximal witness value self-tests a qubit SIC-POVM. Here, we provide all details for the discussion in the main text.

Provide Alice with a random inputx∈ {1,2,3,4}which is associated to a qubit, and Bob with a random inputy∈ {1,2,3}, and consider the following witness

A0SIC= 1 12

X

x,y

P(b=Sx,y|x, y), (C1)

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